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TES: The largest network of teachers in the world
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21.506290 cents
The syntonic or Didymus comma (frequency ratio 81/80) is the smallest superparticular interval which belongs to the 5-limit. Like 16/15, 625/624, 2401/2400 and 4096/4095 it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between 10/9 and 9/8, the product of which is the just major third, 5/4. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. 105edo tempers it out, while 3edo does not.
Tempering out 81/80 gives a tuning for the whole tone which is intermediate between 10/9 and 9/8, and leads to meantone temperament.
Youtube video of "Five senses of 81/80", demonstratory video by Jacob Barton.
According to this interview, Monroe Golden's Incongruity uses just-intonation chord progressions that exploit this comma.
Relations to other Superparticular Ratios
Superparticular ratios, like 81/80, can be expressed as products or quotients of other superparticular ratios. Following is a list of such representations r1 * r2 or r2 / r1 of 81/80, where r1 and r2 are other superparticular ratios.Names in brackets refer to 7-limit meantone extensions, or 11-limit rank three temperaments from the Didymus family that temper out the respective ratios as commas.
External Links
http://en.wikipedia.org/wiki/Syntonic_comma